A nonlinear filtering distributed target tracking method based on variational inference

Abstract

The application belongs to the technical field of target tracking and fusion, and relates to a nonlinear filtering distributed target tracking method based on variational inference. Measurement noise parameters of a conventional nonlinear filtering distributed algorithm are assumed to be known, and the application extends the nonlinear filtering distributed algorithm based on variational inference to a single-target tracking scene under unknown measurement noise parameters. Under the distributed algorithm, weight likelihood parameters of each sensor to the target state tend to be consistent, so that the algorithm has strong robustness, small calculation amount, high flexibility and timeliness, and in combination with the variational inference method, the algorithm can achieve high-precision target tracking in a distributed multi-sensor scene under unknown measurement noise variance. The algorithm has close tracking precision to a distributed particle filtering algorithm with known measurement noise variance, and enhances practicability.

Description

Technical Field

[0001] This invention belongs to the field of target tracking and fusion technology, and relates to a nonlinear filtering distributed target tracking method based on variational inference. Background Technology

[0002] In multi-sensor systems, more accurate and stable tracking can be achieved compared to single-sensor systems. Multi-sensor systems are divided into centralized and distributed systems. Compared to centralized systems, distributed tracking has stronger robustness.

[0003] In single-target tracking scenarios, the Kalman filter algorithm can be accurately applied to target tracking, but it can only track targets with linear trajectories and linear measurements. In contrast, the particle filter algorithm can perform target tracking in nonlinear scenarios.

[0004] However, a significant limitation of the particle filter algorithm is its assumption that all measurement and dynamic model parameters are known prior. In reality, measurement noise statistics are generally time-varying and unknown, making this assumption unreasonable for practical applications. Variational inference, an application of variational methods in statistical inference, can iteratively estimate the local optimum of the latent variable posterior distribution of a probability model within a given variational family by assuming a complex posterior distribution using a simplified distribution. It is frequently used in Bayesian filtering algorithms for the joint posterior estimation of unknown measurement noise parameters and states. Summary of the Invention

[0005] To address this problem, this invention proposes a distributed nonlinear filtering algorithm based on variational inference for situations with unknown measurement noise parameters. Its principle and structure are as follows: Figure 1 As shown, the algorithm first combines traditional filtering algorithms with variational inference to achieve state estimation of multiple distributed sensors under unknown measurement noise conditions. Then, it uses the likelihood consistency algorithm to fuse the state estimation data of multiple sensors to achieve distributed nonlinear filtering based on variational inference.

[0006] The technical solution adopted in this invention is:

[0007] Define the number of sensors in the target tracking scenario as , Time of the first The coordinates of each sensor are represented as follows: , Indicates the target is Position information on the axis ,sensor exist The measured value at time is expressed as The target tracking method includes the following steps:

[0008] S1. Define the motion of the target as conforming to a nonlinear state-space model. The state equations and observation equations of the model are:

[0009]

[0010]

[0011] in, express Time of the first Measurements received by the sensor from the target, process noise Let be zero-mean Gaussian white noise at time t, and Here is its covariance matrix: Let E{·} be the expectation function of the matrix, and the measurement noise is independent of it. Let be the zero-mean Gaussian measurement noise of the k-th sensor at time t, and Here is its covariance matrix: , Let be the measurement function of the k-th sensor. Here is the state transition equation.

[0012] and These are the known nonlinear observation function and the nonlinear state transition function:

[0013]

[0014]

[0015] in The distance traveled at each time step. The time step is set to 1. , Indicates the rotation angle;

[0016] S2. Using the particle filter algorithm, calculate the joint posterior distribution of the state of the k-th sensor and the measurement noise parameters under time-varying unknown conditions:

[0017] definition Time of the first The measurement noise parameters of each sensor are , Time-varying unknowns include:

[0018] (1) Monte Carlo sampling is performed on the particles, and the number of particles is The state probability distribution is obtained as follows:

[0019]

[0020] in Let N be the state value of the i-th particle at time t of the k-th sensor, and N be the number of particles.

[0021] (2) The expected value of the posterior state estimate of sensor k is obtained by the particle filter algorithm principle:

[0022]

[0023]

[0024] in This is the state estimate of the k-th sensor at time t. p represents the weight value of the i-th particle in the k-th sensor at time t. Let S1 be the measurement equation corresponding to the likelihood probability of the measurement value of the k-th sensor at time t. Let be the transition probability of the k-th sensor from time t-1 to time t, corresponding to the transition equation in the measurement equation in S1. The sampled value probability is the transition probability.

[0025] (3) Based on the particle filter algorithm, define get:

[0026]

[0027] Where N(·) is the Gaussian function, Let k be the measurement function of the k-th sensor. Let be the measurement noise variance of the k-th sensor at time t.

[0028] S3, obtained The expression was derived. Following the expression, variational inference is used to calculate... parameter:

[0029] (1) Due to for To satisfy the conjugate prior property, let the variance of be such that . It follows an inverse gamma distribution. The prior distribution is as follows:

[0030]

[0031] in (·) is the inverse gamma function. To measure the dimension of noise variance, Let be the measurement noise variance of the k-th sensor at time t. The measurement value generated by the k-th sensor from time 1 to time t-1. for At time d, the th dimension Standard deviation of measurement noise for each sensor for At time d, the th dimension The prior distribution parameters of the noise variance measured by each sensor. .

[0032] (2) Decompose the prior joint distribution of state and measurement noise parameters into:

[0033]

[0034] Among the particles Obtained by Monte Carlo sampling, therefore... The number of particles;

[0035] (3) The approximate posterior probability using variational inference is:

[0036]

[0037] Where Q(·) is the variational approximation distribution function. for The variational posterior distribution, for The variational approximation posterior function.

[0038] Based on the conjugate prior, the posterior probability density distribution of the measured noise parameters is:

[0039]

[0040] in for Sensor in the d-th dimension at time t The posterior distribution parameters of the measurement noise variance.

[0041] According to the particle filtering algorithm, the posterior probability of a particle is:

[0042]

[0043] (4) Use KL divergence to approximate the variational estimate with the actual value:

[0044]

[0045] in To obtain the minimum KL divergence The function to take the value of.

[0046] Solving the variational approximation posterior yields the coupled equations, and calculating the expectations of the coupled solutions separately yields:

[0047]

[0048]

[0049] in Let be the measurement equation function for the k-th sensor. , It is a constant value.

[0050] Then correspond and From the posterior distribution form, we obtain the variational iterative formula for the parameters:

[0051]

[0052]

[0053]

[0054] in , This is a function to retrieve the value of the d-th dimension.

[0055] (5) To prevent underfitting during iteration, a forgetting factor is introduced. In the iteration with parameter Multiplication indicates the degree of noise fluctuation;

[0056] (6) Substitute the parameters obtained by the variational method into step (1) to obtain the values ​​of k and k for each sensor at time t. Distribution parameter values , and the calculated By combining the particle filter algorithm in S2, the state of the nonlinear system can be determined for each sensor k under unknown measurement noise parameters. The estimate;

[0057] S4. Consider distributed state estimation in a wireless sensor network without a fusion center. Each sensor performs a global estimation task—based on all past and current measurements from all sensors, using only local processing and local communication with its neighbors. A likelihood consensus (LC) algorithm is used to fuse the weight likelihoods of k sensors; specifically: the likelihood consensus algorithm is used to fuse the weight likelihoods of k sensors. Consistency weights are obtained through fusion Define the likelihood probabilities of all sensor measurements as independent of each other, and the joint likelihood function of the sensors as:

[0058]

[0059] in ;

[0060] (1) Global weight likelihood probability:

[0061]

[0062] (2) Because of the sampling points Lacking consistency, so first... Select parameters that are consistent with the data, and then... Make the polynomial assumption:

[0063]

[0064]

[0065] =[ ; ]

[0066] Where m is the order of the polynomial estimate, For the i-th particle at time t of the k-th sensor The sampling part, For the k-th sensor at time t The sample point portion is not included.

[0067] (3) Obtain the new weight likelihood expression:

[0068]

[0069] (4) Define the variable part of each sensor k fusion as follows:

[0070]

[0071]

[0072]

[0073] in , and All of these are intermediate variables fused from the k-th sensor at time t.

[0074] There are global variables:

[0075]

[0076]

[0077]

[0078] in , and All are global intermediate variables fused at time t, and K is the number of sensors.

[0079] (5) The sensor uses only local processing and local communication with its neighbors to obtain the global average in an iterative manner, thereby achieving the summation of all sensors. The iterative formula is as follows:

[0080]

[0081]

[0082] in To find the maximum value function, To retrieve the quantity value of a set, For the number of iterations, , For sensors No. The value of the next iteration. Let k be the set of adjacent sensors. For each iteration of neighbor nodes Weighting;

[0083] (6) After iterating l times, the intermediate mean variable of the fusion is obtained. , , Then substitute it into the global variable formula to obtain the global variable:

[0084]

[0085]

[0086]

[0087] (7) Substitute the global variables into the weight likelihood expression to obtain the global weights of the particle filter:

[0088]

[0089] Where K is the total number of sensors;

[0090] (8) After obtaining the total weight of global consistency Then, the posterior estimate in the joint likelihood function of the sensors is used with the total weight. Instead of using inconsistent data for each sensor. This achieves the goal of data fusion. Furthermore, resampling can be introduced in the particle filtering step to reduce the impact of particle degradation, thereby enabling multi-sensor fusion state estimation under conditions of unknown measurement noise.

[0091] The beneficial effects of this invention are:

[0092] This invention can achieve high-precision target tracking in distributed multi-sensor scenarios with unknown measurement noise variance. Under the distributed algorithm, the weight likelihood parameters of each sensor for the target state tend to be consistent, which has strong robustness, low computational cost, high flexibility and timeliness, and can achieve tracking accuracy close to that of distributed particle filter algorithms with known measurement noise variance. Attached Figure Description

[0093] Figure 1 This is a schematic diagram of the implementation of a distributed nonlinear filtering algorithm based on variational inference.

[0094] Figure 2 Simulation diagram of target tracking using a distributed nonlinear filtering algorithm based on variational inference for a noise measurement with unknown variance.

[0095] Figure 3 Simulation diagram of the mean square error of target tracking coordinates for a distributed nonlinear filtering algorithm based on variational inference with unknown noise measurement variance.

[0096] Figure 4 Simulation diagram of the mean square error of target tracking coordinates for a distributed nonlinear filtering algorithm with known noise measurement variance.

[0097] Figure 5 Simulation diagram of the root mean square error of target tracking coordinates in a distributed nonlinear filtering algorithm that randomly assigns values ​​to the variance of noise measurement. Detailed Implementation

[0098] The present invention will now be described in detail with reference to embodiments:

[0099] In a simulation scenario, five sensors at different locations simultaneously observe a nonlinearly moving target. The target starts from [50m, 20m], with a total movement time T = 50s. The distance traveled in each time step is 80 / Tm, and the turning angle for each movement is pi / T rad. Assume the reconnaissance range... The shaft is -100m to 100m. The axis is -100m to 100m, and the distributed multi-sensor scene coordinates are diag([100,100]')*(rand(2,5)).

[0100] Number of fusion iterations in a distributed manner Set to 30, and assume that each sensor can perform data fusion with its two nearest neighbors, with the measurement covariance set to be time-varying:

[0101] ;

[0102] The system noise covariance is:

[0103] ;

[0104] Distributed particle filter single-target tracking simulation experiments were conducted in three different scenarios. Finally, the average RMSE of the five sensors in the three scenarios was obtained and compared through 100 Monte Carlo experiments.

[0105] Tracking effect:

[0106] To verify the target tracking performance of the distributed nonlinear filtering algorithm based on variational inference, this invention demonstrates the tracking performance of the distributed nonlinear filtering algorithm based on variational inference, the target tracking mean square error of the distributed nonlinear filtering algorithm based on variational inference with unknown measurement noise variance, the distributed nonlinear filtering algorithm with known measurement noise variance, and the distributed nonlinear filtering algorithm with measurement noise variance that does not match the actual variance. Figure 2 Comparing the state estimates from the five sensors with the actual trajectories, it's clear that the state estimates from all five sensors are near the actual trajectories, largely overlapping. This indicates that the tracking algorithm's tracking and fusion effects are good, with the tracked trajectories of the five sensors almost perfectly matching the actual motion trajectories. Figure 3 Separately and Figure 4 and Figure 5 The RMSE comparison further demonstrates that the distributed nonlinear filtering algorithm based on variational inference has good performance in state estimation and fusion of nonlinear systems with unknown measurement noise variance, and also has high tracking accuracy.

Claims

1. A nonlinear filtering distributed target tracking method based on variational inference, defining the number of sensors in the target tracking scenario as... , Time of the first The coordinates of each sensor are represented as follows: , , Indicates the target is Position information on the axis ,sensor exist The measured value at time is expressed as Its characteristics are, The target tracking method includes the following steps: S1. Define the motion of the target as conforming to a nonlinear state-space model. The state equations and observation equations of the model are: ， ， in, express Time of the first Measurements received by the sensor from the target, process noise Let be zero-mean Gaussian white noise at time t, and Here is its covariance matrix: Let E{·} be the expectation function of the matrix, and the measurement noise is independent of it. Let be the zero-mean Gaussian measurement noise of the k-th sensor at time t, and Here is its covariance matrix: , Let be the measurement function of the k-th sensor. Here is the state transition equation; and These are the known nonlinear observation function and the nonlinear state transition function: ， ， in The distance traveled at each time step. For time step, Indicates the rotation angle; S2. Using the particle filter algorithm, calculate the joint posterior distribution of the state of the k-th sensor and the measurement noise parameters under time-varying unknown conditions: definition Time of the first The measurement noise parameters of each sensor are , Time-varying unknowns include: (1) Monte Carlo sampling is performed on the particles, with the number of particles being... The state probability distribution is obtained as follows: ， in Let N be the state value of the i-th particle at time t of the k-th sensor, and N be the number of particles. (2) The expected value of the posterior state estimate of sensor k is obtained by the particle filter algorithm principle: ， ， in This is the state estimate of the k-th sensor at time t. p represents the weight value of the i-th particle in the k-th sensor at time t. Let S1 be the measurement equation corresponding to the likelihood probability of the measurement value of the k-th sensor at time t. Let be the transition probability of the k-th sensor from time t-1 to time t, corresponding to the transition equation in the measurement equation in S1. The probability of the sampled value is the transition probability; (3) Based on the particle filter algorithm, define get: ， Where N(·) is the Gaussian function, Let k be the measurement function of the k-th sensor. Let be the measurement noise variance of the k-th sensor at time t; S3, obtained The expression was derived. Following the expression, variational inference is used to calculate... parameter: (1) Due to for To satisfy the conjugate prior property, let the variance of be such that . It follows an inverse gamma distribution. The prior distribution is as follows: ， in (·) is the inverse gamma function. To measure the dimension of noise variance, Let be the measurement noise variance of the k-th sensor at time t. The measurement value generated by the k-th sensor from time 1 to time t-1. for At time d, the th dimension Standard deviation of measurement noise for each sensor for At time d, the th dimension Prior distribution parameters of the measurement noise variance of each sensor. ; (2) Decompose the prior joint distribution of state and measurement noise parameters into: ， Among the particles Obtained by Monte Carlo sampling, therefore... The number of particles; (3) The approximate posterior probability using variational inference is: ， Where Q(·) is the variational approximation distribution function. for The variational posterior distribution, for The variational approximation posterior function; Based on the conjugate prior, the posterior probability density distribution of the measured noise parameters is: ， in for Sensor at time d in the dimensional The posterior distribution parameters of the measurement noise variance; According to the particle filtering algorithm, the posterior probability of a particle is: ， (4) Use KL divergence to approximate the variational estimate with the actual value: ， in To obtain the minimum KL divergence The function to take the value of; Solving the variational approximation posterior yields the coupled equations, and calculating the expectations of the coupled solutions separately yields: ， ， in Let be the measurement equation function for the k-th sensor. , It is a constant value; Then correspond and From the posterior distribution form, we obtain the variational iterative formula for the parameters: ， ， ， in , This is a function to retrieve the value of the d-th dimension. (5) To prevent underfitting during iteration, a forgetting factor is introduced. In the iteration with parameter Multiplication indicates the degree of noise fluctuation; (6) Substitute the parameters obtained by the variational method into step (1) to obtain the values ​​of k and k for each sensor at time t. Distribution parameter values , and the calculated By combining the particle filter algorithm in S2, the state of the nonlinear system can be determined for each sensor k under unknown measurement noise parameters. The estimate; S4. Use the likelihood consistency algorithm to calculate the weight likelihood of the k sensors. Consistency weights are obtained through fusion Define the likelihood probabilities of all sensor measurements as independent of each other, and the joint likelihood function of the sensors as: ， in ; (1) Global weight likelihood probability: ， (2) Because of the sampling points Lacking consistency, so first... Select parameters that are consistent with the data, and then... Make the polynomial assumption: ， ， =[ ； ]， Where m is the order of the polynomial estimate, For the i-th particle at time t of the k-th sensor The sampling part, For the k-th sensor at time t The sample point portion is not included. (3) Obtain the new weight likelihood expression: ， (4) Define the variable part of the fusion of the k-th sensor as: ， ， ， in , and All of these are intermediate variables fused from the k-th sensor at time t; There are global variables: ， ， ， in , and All are global intermediate variables fused at time t, and K is the number of sensors; (5) The sensor uses only local processing and local communication with its neighbors to obtain the global average in an iterative manner, thereby achieving the summation of all sensors. The iterative formula is as follows: ， ， in To find the maximum value function, To retrieve the quantity value of a set, For the number of iterations, , For the first The first sensor The value of the next iteration. For the first A set of adjacent sensors for each sensor. For each iteration of neighbor nodes Weighting; (6) Iteration After this, the mean intermediate variable after fusion is obtained. , , Then substitute it into the global variable formula to obtain the global variable: ， ， ， (7) Substitute the global variables into the weight likelihood expression to obtain the global weights of the particle filter: ， Where K is the total number of sensors; (8) Obtain the total weight of global consistency Then, the posterior estimate in the joint likelihood function of the sensors is used with the total weight. Instead of using inconsistent data for each sensor. To achieve the goal of data fusion.

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